Question

Prove the identities.

Answer 1

(1)LHS:- 2sec2A- sec4A -2 cosec2A +cosec4A

= 2sec2A - 2cosec2A +cosec4A -sec4A

= 2(1 +tan2A) - 2(1+cot2A) +(1 + cot2A)2 - (1+tan2A)2

= 2 +2tan2A - 2 -2cot2A +1 +cot4A +2cot2A - (1 +tan4A +2tan2A)

= 2tan2A - 2cot2A +1 +cot4A +2 cot2A - 1 -tan4A - 2tan2A

= cot4A - tan4A

= RHS

Hence, proved.


(2)L.H.S=(1+cota- coseca)(1+ tana +seca)

={1+(cos a / sin a) -(1/sin a )} {1 + (sin a / cos a ) + (1/cos a)}

={(sin a + cos a - 1) / sin a } {(sin a +cos a +1) / cos a }

=(1/sin a . cos a ) { (sin a +cos a )^2 - 1}

=(1/ sin a . cos a ) { 1 +2 sin a .cos a -1}

=(1/ sin a . cos a) * (2 sin a . cos a )

=2

=R.H.S